Recent techniques in partial differential equations have led to a solution to the general multidimensional Cauchy problem for nonlinear gradient waves. In a blown-up configuration, Sablé-Tougeron constructs a local solution for a quasilinear hyperbolic system with continuous Cauchy data, in which the first derivatives are discontinuous on a hypersurface. This strong singularity is not so problematic as a rarefaction: The use of Alinhac's para-unknown leads to a tame inequality without loss of derivatives for the iterative scheme.